A conventional water pump used in a water-cooling system for an internal combustion engine or a water-cooled internal combustion engine is disclosed, for example, in Japanese patent application laid-open publication No. 61-001,891 published without examination on Jan. 7, 1986. In detail, as shown in FIGS. 2-4, a conventional water pump 101 includes a pump shaft 102 connected at an end portion thereof with an impeller means 103. The pump shaft 102 is in alignment with a cam shaft 100 which is rotatably mounted in a housing 104 of an internal combustion engine 105. An end portion of the cam shaft 100 is operatively connected to an output shaft 106 of the engine 105 via a timing belt T so that the output rotational movement of the engine 105 is transmitted to the cam shaft 100 while the engine is in operation. An end portion of the pump shaft 102 and an end portion of the cam shaft 100 are connected with each other in male-and-female engaging manner or by a tongue-and-groove joint 100a. Thus, the pump shaft 102 is rotated by the output rotational movement of the engine 105, with the result that the engine 105 is cooled by water circulated by the rotating impeller. However, due to the axial alignment between the cam shaft 100 and the pump shaft 102, the rotational ratio therebetween is determined upon installation of the water pump 101 in the engine. This means that various types of water pumps have to be prepared in accordance with the variations of the engine.